Let \(L_8\) be the set of the 30 binary Lyndon words of length eight, and let \(m\) be the minimum length of a word containing every member of \(L_8\). The current certificates give \[ 49\leq m\leq94. \]

For the lower endpoint, fix a proposed length \(N\). Introduce one Boolean variable for each position of the proposed superstring. For every \(w\in L_8\), assert a disjunction over its \(N-7\) possible starting positions, with eight bit equalities in each disjunct. This formula is satisfiable exactly when a length-\(N\) common superstring exists. Z3 5.0.0 returns `unsat` at \(N=48\), which also rules out every shorter length because a shorter superstring can be padded to length 48.

For the upper endpoint, view each required word as an edge in the order-seven binary de Bruijn graph. The 30 required edges form 11 weak components, each a directed path. Spelling those paths gives blocks of lengths \[ 14,12,10,11,10,8,9,8,9,8,8. \] An exact Held-Karp computation over these 11 blocks finds a shortest block-preserving merge of length 94. One such word is \[ \texttt{0000000111111100000101111100001001111010000110111100010101110001100100101101100011101100110101}. \] The artifact regenerates all targets and checks every containment directly. The interval remains open. In particular, the 94-bit construction is optimal only within the stated block-preserving family.

Fredricksen and Maiorana prove that concatenating, in lexicographic order, the Lyndon words whose lengths divide \(n\) produces a de Bruijn sequence of order \(n\). At \(n=8\), the construction uses Lyndon words of lengths 1, 2, 4, and 8 and covers all 256 binary words of length eight around a cycle.

The present problem asks for a linear word covering only the 30 length-eight Lyndon representatives. Removing the shorter factors from the FKM concatenation does not preserve the theorem's boundary windows, so the construction supplies context and a loose upper bound rather than an exact answer here.

The overlap formulation is standard: for equal-length targets with no containment, an ordering of their occurrences gives a Hamilton path whose arc weight is the longest suffix-prefix overlap. Mucha's shortest-superstring paper uses this reduction to maximum asymmetric traveling-salesperson path. The exact finite computation recorded here instead uses the equivalent order-seven de Bruijn-edge view to expose the 11 required paths.

A length-eight word \(a_1\cdots a_8\) is represented by the directed edge \[ a_1\cdots a_7\longrightarrow a_2\cdots a_8. \] For this target set, each required vertex has at most one required outgoing edge. Starting at every required tail that is not also a required head therefore recovers all 11 maximal paths. Their spells are \[ \begin{gathered} 00000001111111,\ 000001011111,\ 0000100111,\ 00001101111,\\ 0001010111,\ 00011001,\ 000111011,\ 00100101,\\ 001011011,\ 00110101,\ 00111101. \end{gathered} \]

The dynamic program stores the shortest merge ending in block \(i\) for every pair \((S,i)\), where \(S\) is a subset of the 11 blocks. A transition appends block \(j\) after deleting its longest prefix equal to a suffix of block \(i\). Every permutation of the blocks appears in the recurrence, so the value 94 is exact for this restricted family.

The resulting word has SHA-256 digest `3d898b0878cfa058b136a3d1c24244e3154d38a8c30bce767c3fbef686978de0`. Direct substring tests confirm that it contains all 30 generated Lyndon words.

The encoding has 48 Boolean variables \(b_0,\ldots,b_{47}\). For every target word \(w\), it asserts \[ \bigvee_{p=0}^{40}\ \bigwedge_{j=0}^{7}(b_{p+j}=w_j). \] There are 30 target constraints, 1,230 placement terms, and 9,840 bit equalities before internal solver simplification. Any satisfying assignment directly spells a 48-bit common superstring. Conversely, every such superstring supplies a satisfying assignment. The solver returns `unsat`.

This is a solver-assisted exhaustive result. The artifact regenerates the 30 targets and the full formula, so it does not depend on a stored target list or a heuristic search. A separately checkable DRAT or LFSC proof was not retained.